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A study of Difference Equations
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AL-TAHDI UNIVERSITY
FACULTY OF SCIENCE
DEPARTMET OF MATHEMATICS
(A study of Difference Equation on the form)
A DISSERTATION SUBMITTED TO THE DEPARTMENT OF MATHEMATICS IN PARTIAL
FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN
MATHEMATICS​.
BY
FAIZA ATEYA K. ALSSAEDI
SUPERVISOR BY:
DR .NABIL Z .FARID
2007-2008
Chapter(2)
​
Chapter (2)
On the rational recursive sequence​
2.1 Introduction
Qualitive ​analysis difference equations are not only interesting in its own
right, but it can provide insights into their continuous counterparts, namely,
differential equations.
There is a class of nonlinear difference equations, known as the rational
difference equations, each of which consists of the ratio of two polynomials in
the sequence terms in the same form. There has been a lot of work concerning
the global asymptotic behavior of solutions of rational difference equations
[6-7,9-1o,12,13,15-19,21,23 ,24].
Related nonlinear, rational difference equations were investigated in
[5],[8],[14],[22]and[25].
The study of theses equations is quite challenging is in rapid development.
This chapter the boundedness​ , periodicitly ,and the global stability of the equilibrium
Points of the solutions of the following rational difference equations
(2.1)
Where
numbers.
and
є​ (0, ∞) ​and where the initial values
and
are real
2.2 Local stability of the equilibrium point of equation (2.1)
In this section we study the local stability character of the solutions of equation
(2.1).Equation (2.1) has a unique positive equilibrium point and is given by
16
Let f:(0,
)
(0,
) ​→​ (0,
) be the continuous function defined by
Therefore it follows that
Then, we see that
Then the Linearized equation of equation (2.1) about
is
(2.2)
Whose characteristic equation is
(2.3)
​Theorem 2.2.1​ Assume that
<
Then the positive equilibrium point of equation (2.1) is locally asymptotically
stable.
17
Proof: it is follows by Theorem 1.1 that, equation (2.2) is asymptotically stable
if all roots if equation (2.3) lie in the open disc
> 1 that is if
>1
Then,
>1
Thus,
>
Or
>
Then, the proof is complete.
2.3 Boundedness of solutions of equation (2.1)
Here, we study the permanence of equation (2.1).
Theorem 2.3​.​2 Every solutions of equation (2.1) is bounded and persists.
Proof:​ let
be solution of equation (2.1) It follows from equation (2.1)
that
Then​,
for all​
18
Thus,
​
for all
Now, we wish to show that there exists​
<0 such that
for all
Suppose that
,
Will reduce equation (2.1) to the equivalent form
(2.4)
Then,
for all​
Then, we obtain
for all​
19
Thus
for all​
(2.5)
From (2.4) and (2.5), we get
for all​
Therefore, every solution of equation (2.1) is bounded and persists.
2.4 Periodicity of solution of equation​ ​(2.1)
In this section, we study the existence of prime period-two solution of
equation (2.1).
Theorem 2.4.3 Equation (2.1) has positive prime period two solutions if and
only if
>
and
​(a)
Proof:​ First suppose that there exists a prime period two solution
Of equation (2.1).
We see from equation (2.1) that
and
20
Then
​(2.6)
and
(2.7)
Subtracting (2.6) from (2.7) gives
Since,
, it follows that
(2.8)
Also, since
and
are positive, (​
) should be positive.
Again, adding (2.6) and (2.7) yields
(2.9)
It follows by (2.8), (2.9) and the relation
For all
Thus
Then,
21
That
Again, since
Thus,
and
are positive and
<
, we see that
<
(2.10)
Now it is clear from equation (2.8) and equation (2.10) that
and
are the
two positive distinct roots of the quadratic equation
(2.11)
and so
<0
Since
and
are positive,
<
,
which is equivalent to
>
Therefore, inequality (a) holds.
22
Second, suppose that inequality (a) is true.
We will show that equation (2.1) has a prime periodic-two solutions
Assume that
and
We see from inequality (a) that
<
Then,
have the same sign,
and
<
which equivalents to
<
Therefore,
and
are distinct positive real numbers.
Set
,
and
23
We wish to show that
and
It follows from equation (2.1) that
Dividing the denominator and numerator by
gives
Multiplying the denominator and numerator by
24
Gives
,
+
+
25
= .
Similarly as before, one can easily show that
Then it follows by induction that
26
and​
for all
Thus, equation (2.1) has the positive prime period-two solutions
Where
and
are the distinct root of the quadratic equation (2.11) and the
proof is complete.
Lemma 2.4.1 if
=0, then equation (2.1) has no periodic solution of prime
period.
Proof:​ Assume for the sake of contradiction that there exist distinctive positive
real numbers
and
such that
Be two periodic solutions of equation (2.1).then, we see from equation (2.1)
that
and
Which implies
This is a contradiction.
27
2.5 Global stability of equation​ (2.1)
In this section, we investigate the global asymptotic stability of equation
(2.1).
Theorem 2.5.4: The equilibrium point
is a global attractor of equation (2.1) if
one of the following statements holds
and
and
Proof: let
​(2.12)
(2.13)
be a solution of equation (2.1) and again let f be a function
defined by
We will prove (2.12) and (2.13) are similar and will be omitted.
Assume that (2.12) is true, then it is easy to see that the function f is non
decreasing in
and non-increasing in . thus from equation (2.1), we see that
Then
​for all
(2.14)
28
Also, we see from equation (2.1) that
Then
​for all
(2.15)
Then from (2.14) and (2.15), we see that
for all​
It follows by the Method of Full Limiting sequence that there exist solutions
and
Where
of equation (2.1) with
,
It suffices to show that
Now, from equation (2.1) that
29
And so
(2.16)
Similarly, we see from equation (2.1) that
And so
(2.17)
Therefore it follows from equations (2.16) and (2.17)
That
If and only if
Thus
And so
if
Now, we know by (2.12) that
And so it follows that
Therefore
This completes the proof.
30
Chapter(3)
Chapter 3
"On some properties of solution of third order difference
equations"
3.1 Introduction
In this chapter, we will prove some results considering third order
recursive rational difference equations, which particular cases of the
following rational difference equation
The results involve some elements of qualitative theory of difference
equations, namely ,oscillation, periodicity, global attractively and
boundedness nature.
We believe that the results of this chapter are of paramount importance
in their own right and furthermore we believe that these results offer
prototypes towards the development of basic theory of the global behavior of
solution of non-linear difference equations of third order. The techniques and
results of this chapter are also extremely useful in analyzing equations in the
mathematical models of various biological systems and other applications.
3.2 On the d​y​namics of the recursive sequence:
In this section, we investigate the global behavior and the periodic
character of the solutions of the third order recursive difference equation
32
(3. 1)
where the parameters
the initial conditions
and
are nonnegative real numbers and
are nonnegative real number such that
​
By generalizing the results due to [5], the study of such equations are quite
challenging and rewarding and is still in its infancy.
The special cases
We examine the character of solution of (3.1) when one or more of
the parameters of (3.1) are zero.
There are five such equations, namely,
α= 0:
(3.2)
β=0:
(3.3)
(3.4)
33
=0:
(3.5)
(3.6)
In each of the above five equations ,we assume that all parameters in the
(equations are positive. Equation (3.2) is trivial, equations (3.4), (3.5) and (3.6
are linear. Equation (3.3) can also be reduced to a linear difference equation by
the charge of variables
The dynamics of equation (3.1)
We investigate the dynamics of (3.1) under the assumptions that all
parameters in equation are positive and the initial conditions are nonnegative
The change of variables
reduces (3.1) to the difference equation
(3.7)
34
<0 .
Where
Note that
=0 is always an equilibrium point of (3.7),when r>1,(3.7)also
possesses the unique positive equilibrium ​
Theorem 3. 1
The following statements are true:
(1)
The equilibrium point
=0 of (3.7) is locally asymptotically stable if and
only if r<1.
(2)
(3)
The equilibrium point​
=0 of (3.7) is a saddle point if and only if r >1.
When r >1, Then the positive equilibrium point
of (3.7) is
unstable.
Proof​: The Linearized equation of (3.7) about the equilibrium point
So, the characteristic equation of (3.7) about the equilibrium point
is
=0 is
.
And hence, the proof of (1) and (2) follows from The Linearized Stability
Theorem.
For (3), we assume that r >1, Then the linarized equation of (3. 7) about the
35
equilibrium point
has the form
So, the characteristic equation of (3.7) about the equilibrium point
is
(3.8)
It is clear that (3.8) has a root in the interval (-∞,-1) and so,
is
unstable equilibrium point. This completes the proof.
Theorem 3.2​.
Assume that r >1, and let
be a solution of (3.7) such that
and
<
(3.9)
or
Then​
<
=
and
oscillate about​
=
.
​(3. 10)
with semi cycle of length one.
Proof:​ Assume that (3.9) holds [ the case where (3.10) holds is similar and will
be omitted ].
36
Then
​
,
<
and ​
,
Then, the proof follows by induction.
Theorem 3. 3
Assume that r<1, Then the equilibrium point
of
(3.7)is globally
​
asymptotically stable.
Proof:
We know from Theorem (3.1) that the equilibrium point
=0 of
(3.7) is locally asymptotically stable.
So let
be a solution of (3.7). It suffices to show that
Since
​>0
>
​>
So
This complete the proof.
Theorem 3. 4
Assume that r=1, Then (3.7) possesses the prime period-two
solution
37
,​…
….
with
(3. 11)
>0. Furthermore every solution of (3.7) converges to a period two
solution (3.11) with
≥0.
Proof:
Let
​….,
,…
be a period-two solution of (3.7). Then
​ and
so
Which implies that
≤0​.
If​ r<1​, Then it implies that
< 0 or
which is impossible, So​
​r​=1​.
To complete the proof, we assume that r​=1​ and let
of (3.7), Then,
< 0,
be a solution
So the even terms of this solution decrease to a limit (say
≥0), and the
38
odd terms decrease to a limit (say
≥0 ). Thus
​and
,
which implies that
=0
and
=0
This completes the proof.
Theorem 3. 5
Assume that r>1, Then (3.7) possesses unbounded solution.
Proof:​
From theorem 3.2, we can assume without loss of generality
that the solution
​of (3.7)​ is such that
<
and
,Then
>
and ●
<
from which it follows that,
and
Then, the proof is complete.
Remark.
If
=1, the results of [6] follows directly.
39
3.3 On the dynamics of​
​:
The asymptotic stability, periodicity, and trichotomy character of
(3.12)
Was investigated when
initial conditions concerning
and
are non-negative parameters and with
and
to be arbitrary non-negative real
numbers, see[5].
Now, in this section we want to generalize the above results.
consider the third order non-linear rational difference equation
(3.13)
where the parameters
the initial conditions concerning
and
are non-negative real numbers and
and
to
be arbitrary non-negative
​
real numbers. We investigate the global stability and periodic nature of the
solutions of (3.13).
The results which we develop for such equations are also useful in
analyzing the equations in the mathematical models of various biological
systems and other applications [8],[17],[19]and [27].
The special cases
We examine the characters of solution (3.13) when one or more of the
parameters are zero. There are three equations namely:
​
(3.14)
40
(3.15)
​
(3.16)
In each of the above cases we assume that all parameters in the equations
are positive.
Equation (3. 16) are trivial and has a solution
, for all n≥-2. The equations
(3.14) and (3.15) are non-linear third and second order respectively, and the
change of variables
reduce (3.14) and (3.15)to a third and second order linear difference
equation respectively
Now ,We can investigate the dynamics of (3.13) under the assumptions
that all parameters in equation (3.13)are positive and the initial conditions are
non- negative.
The change of variables
reduces (3.13) to the difference equation
​
(3.17)
Where
It is easily to seen that
is always an equilibrium point and when r>1,
we have also an equilibrium point
.
41
An oscillation results​:
Theorem 3.6:
Assume that r>1 and let
be a solution of(3.17) such that either
​and
<
(3.18)
Or
>
and
≥
(3.19)
Then
oscillates about
with
semi-cycle of length one.
​
Proof:
We will assume that (3.18) holds. The case where (3.19) holds is similar
and will be omitted.
Thus we obtain
<
and
≥
and this completes the proof.
Existence of prime period-two solutions
The following theorem shows that (3.17) has a prime-period two
solutions.
Theorem 3.7:
Equation (3.17) has non-negative prime period-two solutions if and only
if either
=0
and
(3.20)
42
or
​
and​ ​
(3.21)
The period two solution must be in the form
(3.22)
Proof​:​ Assume that
​….,
,…
Is a non-negative period two solution of (3. 17).
Then
and
(3. 23)
Hence,
From which, we can see that
​ (3.24)
From (3.23) and (3.24), we get the period two solution in the form
Now, let(3.17) has a period two solutions
Then ,we have either
or
43
If
If​
,Then
​,so​
,so(3. 20)holds.
,
And so
,and(3.21)holds.
This completes the proof.
Local and global stability
It is clearly that
is always an equilibrium solution of (3.17).
Furthermore when r>1, (3.17) also possesses the positive equilibrium
.
Theorem 3.8:
For (3.17), we have the following results.
(1)
The zero equilibrium point is locally asymptotic stable.
(2)
Assume that r>1, then equilibrium point
particular
is unstable. In
is
a saddle point.
​
Proof​:
The linearized equation associated with (3.17) about
has the form
44
So the linearized equation of (3.6) about
and the characteristic equation about
=0 is
=0 is
​,
λ³ = 0.
So, The proof of (1)follows immediately form linearized stability theorem
The linearized equation of (3.6)is a bout
is
and The characteristic equation a bout
is
,with r>1,
Set,
​
f(λ)=​
Then f(1)= -1
< 0 and
(3.25).
f(λ)= ∞, so f(λ) has at least a root in (1,∞)
and the product of the roots of (3.25) is
<1.
​ Hence, there exists a root in a unit disk.
This completes the proof​ .
Theorem 3.9:
Assume that r≤1, then the zero equilibrium of (3.17) is globally
asymptotically stable with basin
​S= [0,∞)
[0,1[²
45
where
(
,
) ≠ (0,1) and (
,
,
)≠ (0,1,0​).
(3.26)
proof:
We know by Theorem 3.81 that
=0 is locally asymptotically
stable
equilibrium point of (3.17), and so it suffices to show that
attractor of (3.6) with basin [0,∞)​
So let
[0,1[² such that (3.26) hold.
be a non-negative solution of (3.17). It is sufficient to show that
Assume that r≤1 and (3.26) hold, we have
.
So, by induction, we have
and
​=0 is a global
46
So , we have
and
are non- increasing sequence and bounded.
Now, suppose that
​and
​
So, we can write
and
​M=L=0​ .
Now, we want to prove that
We must consider the following cases:
(1) If M= 0 and L≠ 0, Then L=1 ⁄r ≥1, which is a contradiction to
is a
non-increasing sequence.
(2) If M≠ 0 and L= 0, Then M =
≥1, which is a contradiction to
non-increasing sequence.
(3) If M ≠0 and L≠ 0, Then we have
M=L=
<0,
is a
Which is a contradiction. So, we must have L=M=0.
This completes the proof​.
Theorem 3.10:
The zero equilibrium point of (3.17) is a globally asymptotically stable
with the basin
47
​S= [0,∞)
[0,
,
,0) and (
[²
where
(
,
)≠(0,
,
)≠(0,
)
(3.27)
Proof:
We know by Theorem 3.8 that
=0 is locally asymptotically stable
equilibrium point of (3.17), and so it suffices to show that
attractor of (3.17) with basin [0,∞)​
So, let
[0,​
[², where (3.27) hold.
be a non-negative solution of (3.17).
It suffices to show that
.
It is easily to seen that
=0 is a global
So, by induction, we have
Which means that
and
are non-increasing and bounded.
48
Now, suppose that
​and​
So, we have
and
Now, we want to prove that M=L= 0. So, we must consider the following
cases:
(1) If M= 0 and L≠ 0, Then L=
, which is a contradiction to
is
a non- increasing sequence.
(2) If M≠ 0 and L= 0, Then M=
a non- increasing sequence.
(3)If M≠ 0 and L≠ 0, then, we have
, which is a contradiction to
is
which is a contradiction.
So, we must have
​L=M=0.
This completes the proof.
Existence of unbounded solutions
In this part, we show that when r>1, (3.17) possesses unbounded
solutions.
49
Theorem 3.11:
Assume that r>1 and x​-1​, x​0
[0,
]. Then (3.17) possessan
unbounded solution.
In particular, every solution of (3.17) which oscillates about equilibrium
, with semicycle of length one is unbounded.
Proof:
We will prove that every solution
of (3.17) which oscillates with
semi-cycles of length one is unbounded (see Theorem. 3.6) for the
existence of solutions which oscillate with semi-cycles of length one).
Let​ r>0 ​ and without loss of generality that
such that
>
and x​2n​>
Then
>
>
and
<
<
from which it follows that
50
​
This completes the proof.
​and
​
51
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54
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‫‪).‬ﯾﻬﺘﻢ ﻣﻮﺿﻮع ﻫﺬه اﻟﺮﺳﺎﻟﺔ ﺑﺪراﺳﺔ ﺧﻮاص ﺣﻠﻮل اﻟﻤﻌﺎدﻻت اﻟﺘﻔﺎﺿﻠﯿﺔ )ﻣﻌﺎدﻻت اﻟﻔﺮوق‬
‫ﻧﺤﻦ ﻧﻌﺘﻘﺪ أن اﻟﻨﺘﺎﺋﺞ اﻟﺘﻲ ﺣﺼﻠﻨﺎ ﻋﻠﯿﻬﺎ ﺣﻮل ﻣﻌﺎدﻻت اﻟﻔﺮوق اﻟﻘﯿﺎﺳﯿﺔ ﻣﻦ اﻟﺮﺗﺒﺔ اﻟﺜﺎﻟﺜﺔ ﻟﻬﺎ أﻫﻤﯿﺔ‬
‫ﺑﺎﻟﻐﺔ ﻓﻲ ﺣﺪ ذاﺗﻬﺎ ﻋﻼوة ﻋﻠﻲ ذﻟﻚ أﻧﻨﺎ ﻧﻌﺘﻘﺪ ﻫﺬه اﻟﻨﺘﺎﺋﺞ ﺗﻘﺪم اﻟﺤﺠﺮ اﻷﺳﺎس ﻧﺤﻮ ﺗﻄﻮر اﻟﻨﻈﺮﯾﺔ اﻷﺳﺎﺳﯿﺔ‬
‫‪.‬ﻟﻠﺴﻠﻮك اﻟﺸﺎﻣﻞ ﻟﺤﻠﻮل ﻣﻌﺎدﻻت اﻟﻔﺮوق اﻟﻐﯿﺮ ﺧﻄﯿﺔ ﻣﻦ اﻟﺮﺗﺒﺔ اﻟﺜﺎﻟﺜﺔ‬
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‫ﺷﻜﺮ وﺗﻘﺪﯾﺮ‬
‫اﻟﺤﻤﺪ ﷲ رب اﻟﻌﺎﻟﻤﯿﻦ‪,‬اﻟﻠﻬﻢ ﻟﻚ اﻟﺤﻤﺪ ﻛﻤﺎ ﯾﻨﺒﻐﻲ ﻟﺠﻼل وﺟﻬﻚ وﻟﻌﻈﯿﻢ ﺳﻠﻄﺎﻧﻚ ‪,‬واﻟﺼﻼة‬
‫‪,‬واﻟﺴﻼم ﻋﻠﻲ ﺳﯿﺪ اﻷوﻟﯿﻦ واﻵﺧﺮﯾﻦ ‪,‬ﺳﯿﺪﻧﺎ ﻣﺤﻤﺪ وﻋﻠﻲ اﻟﻪ وﺻﺤﺒﻪ أﺟﻤﻌﯿﻦ‬
‫‪ :‬وﺑﻌﺪ‬
‫ﻓﺄﻧﻲ أﺗﻘﺪم ﺑﺨﺎﻟﺺ اﻟﺸﻜﺮ واﻟﺘﻘﺪﯾﺮ إﻟﻰ ‪:‬اﻟﺪﻛﺘﻮر ‪/‬ﻧﺒﯿﻞ زﻛﻲ ﻓﺮﯾﺪ ﻋﻠﻲ ﺗﻔﻀﻠﻪ ﺑﺎﻹﺷﺮاف ﻋﻠﻰ‬
‫ﻫﺬه اﻟﺮﺳﺎﻟﺔ ‪,‬وﻋﻠﻰ ﻣﺎ اﻧﻔﻖ ﻣﻦ وﻗﺖ ‪,‬وﺑﺬل ﻣﻦ ﺟﻬﺪ ﻹﺗﻤﺎم ﻫﺬه اﻟﺮﺳﺎﻟﺔ ﻓﻠﻪ ﻣﻦ اﷲ ﺳﺒﺤﺎﻧﻪ‬
‫‪.‬وﺗﻌﺎﻟﻲ ﺧﯿﺮ اﻟﺠﺰاء وأﺣﺴﻨﻪ‬
‫ﻛﻤﺎ ﻻ ﯾﻔﻮﺗﻨﻲ أن أﺗﻮﺟﻪ ﺑﺎﻟﺸﻜﺮ واﻟﺘﻘﺪﯾﺮ إﻟﻰ‪ /‬أﺧﻲ ﻓﺮج و ﻛﻞ ﻣﻦ ﺳﺎﻫﻢ ﻓﻲ أﻋﺪاد ﻫﺬا اﻟﺒﺤﺚ‬
‫‪.‬وﻗﺪم ﻟﻲ اﻟﻨﺼﯿﺤﺔ واﻹرﺷﺎد واﻟﺪﻋﻢ اﻟﻤﺘﻮاﺻﻞ‬
‫ﻓﻼ اﻣﻠﻚ ﻣﺎ أﻗﻮﻟﻪ ﻟﻜﻞ ﻫﺆﻻء إﻻ اﻟﺪﻋﺎء ﻟﻬﻢ ﺑﺎﻟﺨﯿﺮ ‪,‬ﻓﺎﻟﻠﻬﻢ وﻓﻘﻬﻢ ﺟﻤﯿﻌﺎ وﺳﺪد ﺧﻄﺎﻫﻢ‬
‫وﺿﺎﻋﻒ ﻟﻬﻢ اﻷﺟﺮ واﻟﺜﻮاب ‪,‬وﺛﺒﺘﻨﺎ وإﯾﺎﻫﻢ ﺑﺎﻟﻘﻮل اﻟﺜﺎﺑﺖ ﻓﻲ اﻟﺤﯿﺎة اﻟﺪﻧﯿﺎ وﻓﻲ اﻵﺧﺮة ‪,‬اﻧﻚ‬
‫‪.‬ﺳﻤﯿﻊ اﻟﺪﻋﺎء‬
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‫ﺑﺴﻢ اﷲ اﻟﺮﺣﻤﻦ اﻟﺮﺣﯿﻢ‬
‫دﻋﻮاﻫﻢ ﻓﯿﻬﺎ ﺳﺒﺤﺎﻧﻚ اﻟﻠﻬﻢ وﺗﺤﯿﺘﻬﻢ ﻓﯿﻬﺎ ﺳﻼم وأﺧﺮ دﻋﻮاﻫﻢ(​(‬
‫)أن اﻟﺤﻤﺪ ﷲ رب اﻟﻌﺎﻟﻤﯿﻦ(()‪10‬‬
‫ﺻﺪق اﷲ اﻟﻌﻈﯿﻢ‬
‫اﻻﯾﻪ )‪(10‬ﻣﻦ ﺳﻮرة ﯾﻮﻧﺲ‬